Step |
Hyp |
Ref |
Expression |
1 |
|
smndex1ibas.m |
⊢ 𝑀 = ( EndoFMnd ‘ ℕ0 ) |
2 |
|
smndex1ibas.n |
⊢ 𝑁 ∈ ℕ |
3 |
|
smndex1ibas.i |
⊢ 𝐼 = ( 𝑥 ∈ ℕ0 ↦ ( 𝑥 mod 𝑁 ) ) |
4 |
|
smndex1ibas.g |
⊢ 𝐺 = ( 𝑛 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝑥 ∈ ℕ0 ↦ 𝑛 ) ) |
5 |
|
smndex1mgm.b |
⊢ 𝐵 = ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) |
6 |
|
smndex1mgm.s |
⊢ 𝑆 = ( 𝑀 ↾s 𝐵 ) |
7 |
1 2 3 4 5
|
smndex1basss |
⊢ 𝐵 ⊆ ( Base ‘ 𝑀 ) |
8 |
|
dfss |
⊢ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ↔ 𝐵 = ( 𝐵 ∩ ( Base ‘ 𝑀 ) ) ) |
9 |
7 8
|
mpbi |
⊢ 𝐵 = ( 𝐵 ∩ ( Base ‘ 𝑀 ) ) |
10 |
|
snex |
⊢ { 𝐼 } ∈ V |
11 |
|
ovex |
⊢ ( 0 ..^ 𝑁 ) ∈ V |
12 |
|
snex |
⊢ { ( 𝐺 ‘ 𝑛 ) } ∈ V |
13 |
11 12
|
iunex |
⊢ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ∈ V |
14 |
10 13
|
unex |
⊢ ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } ) ∈ V |
15 |
5 14
|
eqeltri |
⊢ 𝐵 ∈ V |
16 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
17 |
6 16
|
ressbas |
⊢ ( 𝐵 ∈ V → ( 𝐵 ∩ ( Base ‘ 𝑀 ) ) = ( Base ‘ 𝑆 ) ) |
18 |
15 17
|
ax-mp |
⊢ ( 𝐵 ∩ ( Base ‘ 𝑀 ) ) = ( Base ‘ 𝑆 ) |
19 |
9 18
|
eqtr2i |
⊢ ( Base ‘ 𝑆 ) = 𝐵 |